FIG. 1 shows a prior art Rotman lens 100, which includes a plurality of equal-length waveguide beam ports 102-1 through 102-n, which couple to parallel plate lens apertures 108-1 through 108-n, respectively. On the opposing side of the parallel plate lens region 114 are a plurality of waveguide apertures 110-1 through 110-m, which are coupled to array port waveguides 104-1 through 104-m which also incorporate the Rotman W parameters, which are incremental per-port delays added to the array port waveguides. Dummy ports 112-1 through 112-p and 106-1 through 106-p couple unusable wave energy which enters from the parallel plate lens region 114 into termination cavities, which minimize wave energy reflected into the parallel plate lens region 114. The beam forming lens 100 may be used bi-directionally, such that for one exemplar transmit application, waveguide RF from a transmitter (not shown) is applied to a power splitter (not shown) and thereafter to a plurality of waveguides and applied to beam port waveguides 102-1 through 102-n, through lens region 114 and waveguide array ports 104-1 through 104-m and thereafter to a transmit antenna. In one examplar receive application, incoming antenna energy is coupled to waveguide array ports 104-1 through 104-m, through parallel plate lens region 114, through waveguide beam ports 102-1 through 102-m, summed (not shown), and delivered to a microwave receiver (not shown).
In an embodiment of the prior art such as U.S. Pat. No. 4,490,723, the Rotman lens 100 of FIG. 1 may be realized using stripline or microstrip conductors, whereby one or more RF conductors are separated by a substrate material having a dielectric constant. Stripline and microstrip transmission lines and lens structures propagate waves in the transverse electromagnetic (TEM) mode. The TEM mode has a phase velocity that is essentially constant with frequency, which results in a formed beam which is largely frequency invariant, which results in the property known as minimum frequency scan, or minimum variation of the formed beam angle with frequency. Prior art U.S. Pat. No. 4,490,723 is one example of this construction. At high operating frequencies, several problems emerge when using stripline or microstrip Rotman lens structures. A first problem is the finite thickness of the dielectric substrate allows the transmission lines formed over the substrate to support higher order wave modes, and the higher order modes propagate at a different phase velocity than the desired TEM mode, thereby causing interference with the desired TEM mode and undesired sidelobes in the radiation pattern. For best performance, the dielectric thickness should be less than 0.1 wavelengths in the dielectric. For example, at an operating frequency of 45.5 Ghz, a wavelength in vacuum is 0.259 inches, which results in a vacuum dielectric thickness of 0.026 inch, and for most substrate dielectrics which have a dielectric constant of approximately 2.2 such as PTFE (PolyTetraFluoroEthylene), a thickness on the order of 0.017 inch, which results in a substrate dielectric structure with undesirably tight feature and etching tolerances. Additionally, many dielectric materials have undesirable mode dependant dielectric constants, and also wave propagation dependant dielectric constants, where in the lens region of a Rotman lens structure, the dielectric constant may depend on the angle of propagation across the planar surface of the lens region.
An alternative to fabricating the Rotman Lens 100 in stripline or microstrip structure is to use a closed waveguide with an air or other dielectric, such as U.S. Pat. No. 6,031,501. The advantage of a waveguide structure is the beam and array waveguides and associated lens structures may be significantly larger and easy to machine and manufacture compared to stripline or microstrip structures, however waveguides support TE modes, and cannot support TEM wave modes. Of the TE modes, TE10 is the lowest mode that can propagate in a rectangular waveguide. For the TE10 mode, the phase velocity Vp is:
  Vp  =      c                  1        -                              (                          λ                              2                ⁢                                  W                  h                                                      )                    2                    
Where:
c=velocity of light;
λ is the free space wavelength
Wh is the height of the waveguide
As can be seen from the formula above, Vp is a function of wavelength λ, which introduces a frequency dependant phase delay producing the result known as frequency scan. The effect of wavelength on Vp can be reduced by maximizing Wh, but this also allows higher mode TE waves to propagate through the waveguide. The TE10 mode is supported by a waveguide with a height Wh of λ/2, TE20 is additionally supported by a waveguide with a height Wh of λ, and TE30 mode is additionally supported when the waveguide height Wh is 3λ/2. It is desired to maximize waveguide height Wh in the lens region, thereby reducing frequency dependant phase velocity which causes frequency scan, while also minimizing the higher modes supported as a consequence of increased Wh. Another desirable outcome of increasing the waveguide height Wh is reduced lens insertion loss.
FIG. 2 shows the geometry of a Rotman lens including lens parameters, which include the four basic lens parameters α, β, f1, γ, where
α is the focal angle shown in FIG. 1;
β is the focal ratio f2/f1 of FIG. 1;
γ is the expansion factor (sin ψ/sin α).
As derived by Hansen, the normalized length W=w/f1 of the waveguide attached to the array element at y=y3 where w is the length of the transmission line to the array port satisfies the following quadratic equation:aW2+bW+c=0
with coefficients a,b,c defined by:
                    a        =                  1          -                                                    (                                  1                  -                  β                                )                            2                                                      (                                  1                  -                                      β                    ⁢                                                                                  ⁢                    cos                    ⁢                                                                                  ⁢                    α                                                  )                            2                                -                                    ξ              2                                      β              2                                                              b        =                              -            2                    +                                    2              ⁢                              ξ                2                                      β                    +                                    2              ⁢                              (                                  1                  -                  β                                )                                                    1              -                              β                ⁢                                                                  ⁢                cos                ⁢                                                                  ⁢                α                                              -                                                    ξ                2                            ⁢                              sin                2                            ⁢                              α                ⁡                                  (                                      1                    -                    β                                    )                                                                                    (                                  1                  -                                      β                    ⁢                                                                                  ⁢                    cos                    ⁢                                                                                  ⁢                    α                                                  )                            2                                                              c        =                              -                          ξ              2                                +                                                    ξ                2                            ⁢                              sin                2                            ⁢              α                                      1              -                              β                ⁢                                                                  ⁢                cos                ⁢                                                                  ⁢                α                                              -                                                    ξ                4                            ⁢                              sin                4                            ⁢              α                                      4              ⁢                                                (                                      1                    -                                          β                      ⁢                                                                                          ⁢                      cos                      ⁢                                                                                          ⁢                      α                                                        )                                2                                                                                      with          ⁢                                          ⁢          ξ                =                                            y              3                        ⁢            γ                                f            1                              
Solving for W for each array port results in a per-array port W distance shown as 202, 204, 206, 208, each of which is computed from the above formulas based on x,y position, and is added to the equal length array port waveguide to arrive at the overall length for each waveguide 104-1 through 104-m of FIG. 1.